5.3 Entropy of d-dimensional extreme black holes

The extremal black holes play a special role in gravitational theory. These black holes are characterized by vanishing Hawking temperature TH, which means that in the metric (165View Equation) the near-horizon expansion in the metric function g(r) starts with the quadratic term (r − r+ )2. Topologically, the true extremal geometry is different from the non-extremal one. Near the horizon the non-extremal static geometry looks like a product of a two-dimensional disk (in the plane (r,τ)) and a (d − 2)-dimensional sphere. Then, the horizon is the center in the polar coordinate system on the disk. Contrary to this, an extremal geometry in the near-horizon limit is a product of a two-dimensional cylinder and a (d − 2)-dimensional sphere. Thus, the horizon in the extremal case is just another boundary rather than a regular inner point, as in the non-extremal geometry. However, one may consider a certain limiting procedure in which one approaches the extremal case staying all the time in the class of non-extremal geometries. This limiting procedure is what we shall call the “extremal limit”. A concrete procedure of this type was suggested by Zaslavsky [226]. One considers a sequence of non-extreme black holes in a cavity at r = rB and finds that there exists a set of data (r+,rB, r− ) such that the limit r+ rB r− → 1, r+ → 1 is well defined. Even if one may have started with a rather general non-extremal metric, the limiting geometry is characterized by very few parameters. In this sense, one may talk about “universality” of the extremal limit. In fact, in the most interesting (and tractable) case the limiting geometry is the product of two-dimensional hyperbolic space H2 with the (d − 2)-dimensional sphere. Since the limiting geometry belongs to the non-extreme class, its classical entropy is proportional to the horizon area in accord with the Bekenstein–Hawking formula. Then, the entanglement entropy of the limiting geometry is a one-loop quantum correction to the classical result. The universality we have just mentioned suggests that this correction possesses a universal behavior in the extreme limit and, since the limiting geometry is rather simple, the limiting entropy can be found explicitly. The latter was indeed shown by Mann and Solodukhin in [172Jump To The Next Citation Point].

5.3.1 Universal extremal limit

Consider a static spherically-symmetric metric in the following form

1 ds2 = g (r )d τ2 + ----dr2 + r2dω2d−2 , (243 ) g(r)
where dω2 d−2 is the metric on the (d − 2)-dimensional unit sphere, describing a non-extreme hole with an outer horizon located at r = r+. However, the analysis can be made for a more general metric, in which −1 gττ ⁄= grr, the limiting geometry is the simplest in the case we consider in Eq. (243View Equation). The function g(r) in Eq. (243View Equation) can be expanded as follows
g(r) = a(r − r+ ) + b(r − r+ )2 + O ((r − r+ )2). (244 )
It is convenient to consider the geodesic distance ∫ −1∕2 l = g dr as a radial coordinate. Retaining the first two terms in Eq. (244View Equation), we find, for r > r+, that
1∕2 (r − r ) = a-sinh2 (lb--) , + b 2 a2 g(lb1∕2) = (--) sinh2(lb1∕2). (245 ) 4b
In order to avoid the appearance of a conical singularity at r = r+, the Euclidean time τ in Eq. (243View Equation) must be compactified with period 4π∕a, which goes to infinity in the extreme limit a → 0. However, rescaling τ → ϕ = τa∕2 yields a new variable ϕ having period 2π. Then, taking into account Eq. (245View Equation), one finds for the metric (243View Equation)
ds2 = 1(sin2xd ϕ2 + dx2) + (r + a-sinh2 x)2dω2 , (246 ) b + b 2 d− 2
where we have introduced the variable x = lb1∕2. To obtain the extremal limit one just takes a → 0. The limiting geometry
1 ( ) ds2 = -- sin2xd ϕ2 + dx2 + r2+d ω2d−2 (247 ) b
is that of the direct product of a 2-dimensional space and a (d − 2)-sphere and is characterized by a pair of dimensional parameters b−1∕2 and r +. The parameter r + sets the radius of the (d − 2)-dimensional sphere, while the parameter −1∕2 b is the curvature radius for the (x,ϕ ) 2-space. Clearly, this two-dimensional space is the negative constant curvature space H2. This is the universality we mentioned above: although the non-extreme geometry is in general described by an infinite number of parameters associated with the determining function g(r), the geometry in the extreme limit depends only on two parameters b and r+. Note that the coordinate r is inadequate for describing the extremal limit (247View Equation) since the coordinate transformation (245View Equation) is singular when a → 0. The limiting metric (247View Equation) is characterized by a finite temperature, determined by the 2π periodicity in angular coordinate ϕ.

The limiting geometry (247View Equation) is that of a direct product H2 × S2 of 2d hyperbolic space H2 with radius −1∕2 l = b and a 2D sphere S2 with radius l1 = r+. It is worth noting that the limiting geometry (247View Equation) precisely merges near the horizon with the geometry of the original metric (243View Equation) in the sense that all the curvature tensors for both metrics coincide. This is in contrast with, say, the situation in which the Rindler metric is considered to approximate the geometry of a non-extreme black hole: the curvatures of both spaces do not merge in general.

For a special type of extremal black holes l = l1, the limiting geometry is characterized by just one dimensionful parameter. This is the case for the Reissner–Nordström black hole in four dimensions. The limiting extreme geometry in this case is the well-known Bertotti–Robinson space characterized by just one parameter r+. This space has remarkable properties in the context of supergravity theory that are not the subject of the present review.

5.3.2 Entanglement entropy in the extremal limit

Consider now a scalar field propagating on the background of the limiting geometry (247View Equation) and described by the operator

2 𝒟 = − (∇ + X ), X = − ξR (d), (248 )
where R(d) is the Ricci scalar. For the metric (247View Equation) characterized by two dimensionful parameters l and l 1, one has that R = − 2∕l2 + (d − 2 )(d − 3)∕l2 (d) 1. For a d-dimensional conformally-coupled scalar field we have -d−2- ξ = 4(d−1). In this case
Xconf = − (d −-2)(d −-4)+ (d-−-2)-(1∕l2 − 1 ∕l2) . (249 ) 4l2 2(d − 1) 1

The calculation of the respective entanglement entropy goes along the same lines as before. First, one allows the coordinate ϕ, which plays the role of the Euclidean time, to have period 2πα. For α ⁄= 1 the metric (247View Equation) then describes the space α α E = H 2 × Sd−2, where α H 2 is the hyperbolic space coinciding with H2 everywhere except the point x = 0, where it has a conical singularity with an angular deficit δ = 2π (1 − α ). The heat kernel of the Laplace operator ∇2 on E α is given by the product

KE α(z,z′,s) = KH α(x,x ′,ϕ,ϕ ′,s) KSd −2(𝜃, 𝜃′,φ, φ′,s) 2
where KH α2 and KS2 are the heat kernels, of the Laplace operator on α H 2 and Sd−2 respectively. The effective action reads
1 ∫ ∞ ds We ff[E α] = −-- 2 ---TrKH α2 T rKSd−2eXs , (250 ) 2 𝜖 s
where 𝜖 is a UV cut-off. On spaces with constant curvature the heat kernel function is known explicitly [34]. In particular, on a 2D space H2 of negative constant curvature, the heat kernel has the following integral representation:
√ -- 1 2e− ¯s∕4 ∫ ∞ dyye −y2∕4¯s KH2 (z,z′,s) = -2-----3∕2 √----------------, (251 ) l (4π ¯s) σ coshy − cosh σ
where ¯s = sl−2. In Eq. (251View Equation) σ is the geodesic distance between the points on H2. Between two points (x,ϕ) and (x, ϕ + Δ ϕ) the geodesic distance is given by sinh2 σ = sinh2 x sin2 Δ-ϕ 2 2. The heat kernel on the conical hyperbolic space α H 2 can be obtained from (251View Equation) by applying the Sommerfeld formula (22View Equation). Skipping the technical details, available in [172Jump To The Next Citation Point], let us just quote the result for the trace
e− ¯s∕4 2 TrKH α2 = αTrKH2 + (1 − α)(4π-¯s)1∕2kH (¯s) + O(1 − α ) , ∫ ( ) ∞ cosh-y- ---2y-- − y2∕¯s kH(¯s) = 0 dy sinh2 y 1 − sinh 2y e , (252 )
where ¯s = s∕l2.

Let us denote Θd−2(s) = TrKSd −2(s) the trace of the heat kernel of the Laplace operator − ∇2 on a (d − 2)-dimensional sphere of unit radius. The entanglement entropy in the extremal limit then takes the form [172Jump To The Next Citation Point, 206Jump To The Next Citation Point]

∫ ∞ ( 2) Sext = -1√--- -ds-kH(s)Θd −2 sl- e−s∕4esXl2 . (253 ) 4 π 𝜖2∕l2 s3∕2 l21
The function kH(s) has the following small-s expansion
√ ---( 1 1 17 29 1181 1393481 763967 ) kH (s ) = πs --− ---s + ----s2 − -----s3 + -------s4 − ----------s5 + ----------s6 + ..(254 ) 3 20 1120 4480 337920 615014400 447283200
The trace of the heat kernel on a sphere is known in some implicit form. However, for our purposes a representation in a form of an expansion is more useful,
Ω ( ∑∞ ) Θd −2(s) = -----d(−d2−2)∕2- 1 + (d − 2)(d − 3) a2nsn , (255 ) (4πs) n=1
where Ω = 2π(d−1)∕2-- d−2 Γ ((d−1)∕2) is the area of a unit radius sphere S d−2. The first few coefficients in this expansion can be calculated using the results collected in [219Jump To The Next Citation Point],
1- (5d2-−-27d-+-40)- (35d4-−-392d3-+-1699d2-−-3322d-+--2520)- a2 = 6 , a4 = 360 , a6 = 45360 . (256 )

We shall consider some particular cases.

d=4.
The entanglement entropy in the extreme limit is

2 ( ) 2 Sd=4 = -l1--+ s0ln 𝜖 + s l1 , s0 = 1-(6ξ − 1) + 1--l1(1 − 5ξ) , (257 ) 12𝜖 l l 18 15 l2
where s(l1) l is the UV finite part of the entropy. For minimal coupling (ξ = 0) this result was obtained in [172]. The first term in Eq. (257View Equation) is proportional to the horizon area 2 A = 4πl1, while the second term is a logarithmic correction to the area law. For conformal coupling ξ = 1∕6, the logarithmic term is
conf -1-l21- s0 = 90 l2 . (258 )

d=5.
The entropy is

√ -- √ -- ( ) πl31 π ( 2 2 2 2) l1 l1 Sd=5 = ----3 + ---- (2l1 − 5l ) + 10ξ(3l − l1) -2-+ s -- . (259 ) 72𝜖 120 l 𝜖 l
To simplify the expressions in higher dimensions we consider only the case of the conformal coupling ξ = -d−-2- 4(d−1).

The entropy takes the form:

d=6.

( ) --l41--- --1-l21(4l2 −-5l12)- 𝜖- l- Sd=6 = 144𝜖4 + 180 l2 𝜖2 + s0 ln l + s l , ( )1 1 --1--- l41 l21- s0 = − 18900 1068 l4 − 1680 l2 + 637 . (260 )

d=7.

√-- √ -- πl51 7 π (25l2 − 32l21)l31 Sd=7 = -----5 + ------------2-3----- 384 𝜖√ -- 34560 4 l 𝜖 22 4 ( ) + ----π---(70592l-1 −-109760l-1l-+-40635l-)l1+ s l1- . (261 ) 1935360 l5𝜖 l

d=8.

( ) S = s6 + s4 + s2 + s ln 𝜖 + s l1- , (262 ) d=8 𝜖6 𝜖4 𝜖2 0 l l6 1 l4(− 209l2 + 160l2) s6 = --1--, s4 = -------1------1--------- , 2160 75600 l6 ---1---l21(8753l41 −-13376l2l21 +-4875l4) s2 = 352800 l6 , 1 (1102263l6 − 2520864l2l4+ 1833975l4l2 − 413120l6 ) s0 = --------- ---------1-------------1-------------1------------. (263 ) 11113200 l6
Two examples of the extreme geometry are of particular interest.

Entanglement entropy of the round sphere in Minkowski spacetime.
Consider a sphere of radius R in flat Minkowski spacetime. One can choose a spherical coordinate system (τ,r,𝜃i) so that the surface Σ is defined as τ = 0 and r = R, and variables 𝜃i , i = 1,..,d − 2 are the angular coordinates on Σ. The d-metric reads

2 2 2 2 2 ds = dτ + dr + r dωd−2 , (264 )
where dω2 d−2 is a metric on (d − 2) sphere of unit radius. Metric (264View Equation) is conformal to the metric
R2 ds2ext = -2-(dτ2 + dr2) + R2d ω2d− 2, (265 ) r
which describes the product of two-dimensional hyperbolic space H2 with coordinates (τ,r) and the sphere Sd− 2. Note that both spaces, H2 and Sd− 2, have the same radius R. Metric (265View Equation) describes the spacetime, which appears in the extremal limit of a d-dimensional static black hole. In the hyperbolic space H2 we can choose a polar coordinate system (ρ,ϕ) with its center at point r = R,
R R sinh ρ sin ϕ r = --------------------, τ = --------------------, (266 ) cosh ρ − sinh ρ cosϕ cosh ρ − sinh ρ cosϕ
(for small ρ one has that r = R + ρcos ϕ and τ = ρ sin ϕ as in the polar system in flat spacetime) so that the metric takes the form
2 2 2 2 2 2 2 dsext = R (dρ + sinh ρd ϕ ) + R dω d−2. (267 )
In this coordinate system the surface Σ is defined by the condition ρ = 0. In the entanglement entropy of a conformally-coupled scalar field the logarithmic term s0 is conformally invariant. Therefore, it is the same [206] for the entropy of a round sphere of radius R in Minkowski spacetime and in the extreme limiting geometry (267View Equation). In various dimensions s0 can be obtained from the results (258View Equation) – (262View Equation) by setting l = l1 = R. One finds s0 = -1 90 in d = 4, s0 = − 1-- 756 in d = 6 and s0 = -23-- 113400 in d = 8. For d > 4 the logarithmic term in the entropy of a round sphere has been calculated by Casini and Huerta [38Jump To The Next Citation Point] directly in Minkowski spacetime. They have obtained s0 in all even dimensions up to d = 14. Subsequently, Dowker [72Jump To The Next Citation Point] has extended this result to d = 16 and d = 18. In arbitrary dimension d the logarithmic term s0 can be expressed in terms of Bernulli numbers as is shown in [38] and [72]

Entanglement entropy of the extreme Reissner–Nordström black hole in d dimensions.
As was shown by Myers and Perry [180Jump To The Next Citation Point] a generalization of the Reissner–Nordström solution to higher dimension d > 4 is given by Eq. (243View Equation) with

(rd−3-−-rd+−3)(rd−3 −-rd−−3) g(r) = r2(d− 3) . (268 )
In the extreme limit r+ → r−. Expanding Eq. (268View Equation) near the horizon one finds, in this limit, that b = (d − 3)2∕r2+. Thus, this extreme geometry is characterized by values of radii l = r+∕(d − 3) and l1 = r+. In dimension d = 4 we have l = l1 as in the case considered above. In dimension d > 4 the two radii are different, l ⁄= l1. For the conformal coupling the values of logarithmic term s0 in various dimensions are presented in Table 1.


Table 1: Coefficients of the logarithmic term in the entanglement entropy of an extreme Reissner–Nordström black hole.
d 4 6 8
s0 1 90 2881 − 756- 1569275563 -1111320--


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